Equilibrium properties of short field-reversed configurations

Equilibrium properties of short field-reversed configurations
Some features of short field-reversed configuration (FRC) equilibria relevant to transport and stability are studied numerically and analytically. It is shown that magnetic field curvature effects significantly increase the FRC magnetization for plasma elongations epsilon is less than or equal to 4.
Equilibrium properties of short field-reversed configurations M. Tuszewski Los Alamos National Laboratory, Los Alamos, New Mexico 87545 R. I.. Spencer Brigham Young University, Provo, Utah 84602 (Received 28 January 1986; accepted 12 August 1986) Some features of short field-reversed configuration (FRC) equilibria relevant to transport and stability are studied numerically and analytically. It is shown that magnetic field curvature effects significantly increase the FRC magnetization for plasma elongations E<4. (1) I. INTRODUCTION A field-reversed configuration (FRC) is an elongated compact toroid without a toroidal field. ! These plasmas are formed in prolate theta pinches with a reversed bias field. In most experiments, PRC's have aspect ratios E (the ratio of the separatrix length to the diameter) sufficiently large so that magnetic field curvature is negligible over the central portion of the separatrix. For such cases, assuming no plasma pressure outside the separatrix and an infinitely long flux conserver ofuniform radius, an axial equilibrium constraint (first(fd3e)ri=ve-dlblyr.B-apr2nersd) isro=bt1aine-dI-!X:-;, 0 PM 2 2 where P= plpM is the plasma pressure normalized to the maximum pressure at the field null and where Xs = rslrw is the ratio of midplane separatrlx radius to coil radius. The average beta condition ofEq. (1) is a stringent constraint on the FRC radial pressure profile. It has been found to be consistent with the available experimental data,2 and has often been used in equilibrium and transport studies. Various corrections to Eq. (1) have been derived to account for finite particle orbits,3 plasma pressure on open field lines,4 end mirrors,4,s and toroidal field.4 Magnetic field curvature may significantly modify Eq. ( 1) if PRC equilibria are produced with sufficiently small values of E. Such small values of E could result from separatrix length shrinkage,6 which occurs whenever energy losses exceed po10idal flux loss. Such short equilibria are currently being studied experimentally. In the VLX device7 PRC confinement in a coil ofaspect ratio (ratio ofuniform coil region to coil diameter) variable in the range 1-4 is being investigated, and in the FRX-C device8 short PRC equilibria with E in the range 3-4 have been recently obtained with a coil aspect ratio of2.6. Even shorter FRC equilibria may be obtained in a modified version of this device9 with a coil aspect ratio of about 2. II. EQUILIBRIUM COMPUTATIONS In this paper, we compute the effect offield curvature on PRC equilibria. Short equilibria were obtained in straight cylinder geometry (no end mirrors) by using the same code as in previous work.4 In each of the cases discussed in this paper, the equilibrium length was adjusted by changing the total toroidal current; Ps (the pressure at the separatrix normalized to the pressure at the field null) and Xs were held fixed by adjusting the pressure profile, p (l/J). For each equilibrium, the integral given in Eq. (1) was computed and the ratio (f3 )I (1 - ! x;) was obtained. These ratios are shown as functions ofE in Fig. 1for various values ofXs andps' The open circles correspond to x. = 0.4 and P. = 0.6 and the solid circles correspond to Xs = 0.6 and Ps = 0.3. Also shown with a triangle in Fig. 1 is the analytical spherical Hill's vortex equilibrium (P. = 0) which is a solution in a uniform flux conserverlO in the limit ofx. -.0. For this case, onehas (f3) = at E = 1. Theequilibria represented in Fig. 1 by open circles do not appear to map towards the Hill's vortex value, presumably because of the finite values of X s andPs .Fixing Ps and reducing E produced rather odd-looking equilibria with a lot of magnetofluid outside the separatrix. To see what happens when this fluid is removed, two additional equilibria (open circles with a bar) were computed with x. = 0.4 and P. = 0.1. These equilibria have lower values of (f3) than those with x. = 0.4 and P. = 0.6 and better extrapolate toward the Hill's vortex value. The results -(\l N'w f!. x t: Ii 0.5 V 4 E FIG. 1. Computed values of (fJ ) relative to elongated values as functions of . The open circles are forx. = 0.4and,8. = 0.6, the open circles with a bar are for x. = 0.4 and P, = 0.1, and the solid circles are for x. = 0.6 and ,8. = 0.3. The triangle is for the Hill's vortex case with x, =,8. = 0, and the solid line is the analytical expression of Eq. (3) with x. = 0.5. For each case, the value (fJ) u' the average of beta over the separatrix volume, normalized to I !X , is shown with a cross. 3711 Phys. Fluids 29 (11), November 1986 0031-9171/86/113711-04$01.90 @ 1986 American Institute of Physics 3711 FIG. 2. Computed radial pressure profiles relative to the maximum pressure PM for x, =0.4 and (a) E = 3.9, (b) E = 2.7, and Copyright E = 1.4. The dotted line is an approximate sharp-boundary model for curve Copyright. 1 (2) lrw B 2 -- ---21TrdrI . 11'':; 0 21l0PM z>O The first integral on the right-hand side of Eq. (2) can be evaluated using B 2/2JLOPM = I-p/PM forr<rs andB 2/ IlloPM = 1 - {j(p = 0) for r> rs ' In the second integral we use flux conservation, B Iz>o =Bw (1 - x;), and our modified radial pressure balance, B /2JLoPM = 1 - 8. With these substitutions, and some rearrangement, Eq. (2) can be written (f3) = l-x;12 - (8/2)(1-x;). (3) We now make an estimate of the quantity 8. With field-line curvature included, radial pressure balance, in the midplane, becomes ty is quite large and localized near the separatrix. This is a rather drastic approximation, but since the quantity ofinterest, (/3 ), is an integrated quantity, it is not as bad as might be expected. Note that as E becomes large, this model is trivially exact with {j = O. With this model the integral for (f3 ) in Eq. ( 1) can be evaluated, following the derivation outlined in previous work. 1 We start with Eq. (A7) of Ref. 1. When normalized to 11'':;PM' this equation becomes 1 lrw B 2 (f3) = - --211'7 dr I 11'':; 0 21l0PM z = 0 B 2 JB 2 p+-= --dr, 2110 lloP whereP is the radius ofcurvature of the magnetic field lines. To a good approximation, FRCseparatrices 's (z) have been shown experimentally2.4 to be elliptical in shape, so that at r = rs and z = 0 one obtains p = - rs c. Assuming that the integrated curvature factor {j is proportional to E- 2, we write 8 = a/c. The coefficient a can now be estimated by computing it for the simplest finite-length FRC equilibrium, the Hill's vortex. We do so by requiring Eq. (3) to hold for this equilibrium. With (/3) = 2/3, X s <1, and E = 1, we obtain {j = a = 2/3. We therefore assume {j = 2I3c and rewrite Eq. (3) as (/3) = 1 -x;12 - (1- x;)/3c. (4) The solid line on Fig. 1corresponds to Eq. (4) withxs = 0.5. This curve fits well the numerical data over the entire range of E. Equation (4) provides a simple approximation to (f3) for arbitrary aspect ratio E that can easily be evaluated experimentally. The factor 1 - x; that appears in Eq. (4) implies decreased curvature effects with larger values ofxs ' in qualitative agreement with the numerical data ofFig. 1. This can probably be attributed to the effect ofthe flux conserving wall; at large xs ' the plasma is pushed against the straight wall, and tends to become flat. The numerical data of Fig. 1 also suggest that curvature effects are lessened with increasing/ 3s; this effect is not included in Eq. (4). The FRG total temperature T is commonly inferred from radial pressure balance once B w at radius r = rw and nM at radius, = 's/21/2 are determined from excluded flux measurements and from side-on interferometry. Therefore, the expressions PM = nMT= B l2llo(1-{j) and {j = 21 o in Fig. I show that curvature effects are negligible for E 4 but rapidly become significant as E approaches unity. Comparison between the various cases suggests that curvature effects are somewhat reduced by larger values ofXs and /3s . For highly elongated FRC's, it is normally assumed that (f3 ), the midplane-averaged beta, is the same as (f3) v' the volume-averaged beta. Because (f3) is easily determined experimentally, it is often used, for the case ofhighly elongated FRC's, as a substitute for the more physical quantity (f3 ) v However, as E approaches 1, the values of (f3) v can diverge significantly from the values of(f3 ). This is illustrated in Fig. 1 where the values of (f3) v' normalized to 1 - x;, are shown with crosses for each computed case. Note that both (/3) and (/3 ) v decrease with decreasing E. However, the ratio (f3 ) v / (f3 ) decreases from O.95 to 0.69 as E decreases from 6 to 1. Hence, there is no obvious relation between (f3 ) and (/3 ) v ; furthermore. because ofpossible wide variations in the axial distribution of flux surfaces with variations inP ( t/!),4a simple relation between (f3 ) and (f3 )" may be quite difficult to find. III. ANALYTIC MODEL In order to gain more insight, an attempt was made to fit a simple analytical model for curvature effects. Thenumerical equilibria revealed midplane radial profiles of total pressure (p +B 2/2JLO) with a drop largely localized in the vicinity of the separatrix. Three examples are given in Fig. 2 for various values ofE and for Xs = 0.4. These examples suggest an approximate model with uniform regions oftotal pressure inside and outside the separatrix. An example for curve Copyright is sketched with a dotted line on Fig. 2. We assumeP +B 2/ 2JLo =PM inside the separatrix andp +B 2/2p,0 = PM (I - {j) outside the separatrix. Note that the discontinuity in total pressure in this model implies that B VB is proportional to a delta function at the separatrix, i.e., implies that this quanti- (a) lb) a. ..... 0 ::t. (e) C\l -. C\I CO as + eo 3712 Phys. Fluids, Vol. 29, No. 11, November 1986 M. Tuszewski and R. L. Spencer 3712 (5) ACKNOWLEDGMENT This research was supported by the United States Department of Energy. FIG. 3. Computed values ofs for short equilibria, normalized to the corresponding values off.> 1 for highly elongated equilibria (E = 6.3 for x, = 0.4 and E = 4.5 for x, = 0.6) are displayed as functions of E. The various cases are the same as those of Fig. 1. The solid lines correspond to the relation s-;- [I +61X:(l - 6) ]1/2, normalized to the same expression evaluated forE = 6.3 (x, = 0.4, uppercurve) and forE = 4.5 (x, = 0.6, lowercurve). 4 5 6 E 2 3 With 8 = 213 , this relation is shown with solid lines in Fig. 3 for Xs = 0.4 and for Xs = 0.6. The quantity 8,,> I is a value of8 corresponding to E = 6.3 for X s = 0.4 and to E = 4.5 for X s = 0.6. We observe from Fig. 3 that the numerical values ofnormalized Svalues are indeed in good agreement with the approximate upper-bound scaling. The relation (J-Bwx; [1 +l5/x;(1 - tJ>] 1/2 (6) may therefore be a useful way to use simple magnetic measurements in present experiments to estimate the flux confinement times ofshort FRC's. Similarly, the values of (f3 >" shown in Fig. I could perhaps be used to estimate the energy confinement times of short FRC's from the relation E- (f3 >v VB /(l -15),whereEistheplasmaenergyand V is the separatrix volume. IV. CONCLUSIONS In conclusion, some features of short FRC equilibria have been studied numerically. Specifically, it is shown that magnetic field curvature significantly increases the FRC magnetization for elongations E<4. This should improve FRC transport and might decrease the effectiveness ofkinetic stabilization. The transport parameter (f3 >is computed and a simple analytical approximation for it is derived. The relative increases ofthe stability parametersand ofthe FRC trapped flux (J over one-dimensional estimates are evaluated. It is found that curvature effects are somewhat reduced by increased values of X s and f3s. ...... len len 1.5 3 provide a temperature estimate for cases with low values ofE. The values of (f3 >given by Eq. (4) can be used to extend existing one-dimensional transport models, such as the one of Ref. 3, to the range E-2-4. In this range, end effects that reducell.'2confinementtimesbyafactor /(+ Vremain small while the decreased (f3 > (and therefore decreased pressure gradients) should yield improved cross-field transport. A parameter of importance for FRC stability is -s= ir'r-d-r, R rsPi the approximate number13 of ion gyroradii, Pi' between the field null R and the separatrix rs A recent kinetic theory14 of the internal tilt mode suggests sufficiently small growth rates for present experiments with $< 1-2 to be stable, but observable instability in future experiments with 8> 3-4. If 8 has the same significance with regard to stability for these short equilibria, then such equilibria might represent a simpler way to study the tilt mode. Their shorter length, however, could substantially alter the eigenfunction (for instance, by changing it from an axial displacement to a rigid rotation, or by changing the important instability from an internal tilt mode to an external tilt mode, or by changing it into some combination of all of these) making it difficult to compare with the physics oflong equilibria. For cases with small values ofE, the usual one-dimensional estimates'2 of 8 become inaccurate. Since 8 is proportional to (J/rs , where (J is the FRC poloidal fiux, and where we have assumed constant ion temperature, one can readily compute the relative increase of 8 over its elongated value, for a given value of X s ' by calculating (J ratios. The ratios S/8,,>1 = (J/(J,,> I from the numerical data of Fig. 1 are shown in Fig. 3. For the cases with Xs = 0.4 we used an equilibrium with E = 6.3 to obtain (J,,> I' and for Xs = 0.6 we used an equilibrium with E= 4.5. We observe from Fig. 3 significant relative increases of s for E<3 that correspond to the enhanced magnetization suggested in Fig. I by lower values of (f3 ). The reduction of curvature effects with larger values ofX s and f3s noted previously is also apparent in Fig. 3. For elongated FRC's, Eq. (1) has been usedl 6 to provide bounds on the FRC poloidal flux (J. For short FRC's, Eq. (4) can be used in a similar way to obtain I.:B X (l ) 11' III "'4 + x;(1 _ tJ) x; ( tJ )112 <(J<11' Bw 1+ 2 2 xs (1- tJ) The bounds for (J given in Eq. (5) are higher than those of elongated FRC's (corresponding to 8 = 0). It is often assumed6 that the elongated FRC's of present experiments have values of(J much closer to the upper bound than to the lower bound. This is still true for the short numerical equilibria of this paper. Neglecting variations in Bw ' the upper bound of Eq. (5) implies (J [1 +tJ/x;(l-8)]1/2 (J,,>I = [1 +tJ">I/X;O - tJ">l >] 1/2 . 3713 Phys. Fluids, Vol. 29, No. 11, November 1986 M. Tuszewski and R. L. Spencer 3713 IW. T. Armstrong, R. K. Linford, J. Lipson, D. A. J1latts, and E. G. Sherwood, Phys. Fluids 24, 2068 (1981). 2M. Tuszewski, Plasma Phys. 26, 991 (1984). 3M. Tuszewski and R. K. Linford, Phys. Fluids 25, 765 (1982). 4R. L. Spencer and M. Tuszewski, Phys. Fluids 28, 1810 (1985). 'K. Suzuki and S. Hamada, J. Phys. Soc. Jpn. 53,16 (1984). 6M. Tuszewski, W. T. Armstrong, R. R. Bartsch, R. E. Chrien, J.Copyright Cochrane, Jr., R. W. Kewish, Jr., P. K1ingner, R. K. Linford, K. F. McKenna, D. J. Rej, E. G. Sherwood, and R. E. Siemon, Phys. Fluids 25, 1696 (1982). 'J. A. Wilson, Z. A. Pietrzyk, and D. E. Lotz, Bull. Am. Phys. Soc. 29, 1360 (1984). 8M. Tuszewski, in Proceedings of the 7th Symposium on the Physics and Technology ofCompact Toroids, Santa Fe, 1985 (Los Alamos National Laboratory, Los Alamos, NM, 1986), p. 142. . Tuszewski, R. E. Chrien, E. J. Yavomik, W. T. Armstrong, W. Hugrass, R. K. Linford, K. F. McKenna, D. J. Rej, and R. E. Siemon, in Proceedings ofthe 7th Symposium on the Phpics and Technology ofCompact Toroids, Santa Fe, 1985 (Los Alamos National Laboratory, Los Alamos, NM, 1986), p. IS2. 10Jt. L. Spencer and D. W. Hewett, Phys. Fluids 15, 1365 (1982). I1S. P. Auerbach and W.Copyright Condit, Nucl. Fusion 21, 927 (1981). 12K. Nguyen and T. Kammash, Plasma Phys. 24,177 (1982). 13J. T. Slough, A. L. Hotrman, R. D. Milroy, D. G. Harding, and L.Copyright Steinhauer, Nucl. Fusion 24, 1537 (1984). 1 4J. L. Schwarzmeier, H. R. Lewis, D.Copyright Barnes, and C. E. Seyler, Bull. Am. Phys. Soc. 29,1327 (1984). 3714 Phys. Fluids, Vol. 29. No. 11, November 1986 M. Tuszewski and R. L. Spencer 3714

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Some features of short field-reversed configuration (FRC) equilibria relevant to transport and stability are studied numerically and analytically. It is shown that magnetic field curvature effects significantly increase the FRC magnetization for plasma elongations epsilon is less than or equal to 4.

(c) 1986 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in The Journal of Chemical Physics and may be found at http://dx.doi.org/10.1063/1.865803;

Equilibrium properties of short field-reversed configurations
Some features of short field-reversed configuration (FRC) equilibria relevant to transport and stability are studied numerically and analytically. It is shown that magnetic field curvature effects significantly increase the FRC magnetization for plasma elongations epsilon is less than or equal to 4.
Equilibrium properties of short field-reversed configurations M. Tuszewski Los Alamos National Laboratory, Los Alamos, New Mexico 87545 R. I.. Spencer Brigham Young University, Provo, Utah 84602 (Received 28 January 1986; accepted 12 August 1986) Some features of short field-reversed configuration (FRC) equilibria relevant to transport and stability are studied numerically and analytically. It is shown that magnetic field curvature effects significantly increase the FRC magnetization for plasma elongations E<4. (1) I. INTRODUCTION A field-reversed configuration (FRC) is an elongated compact toroid without a toroidal field. ! These plasmas are formed in prolate theta pinches with a reversed bias field. In most experiments, PRC's have aspect ratios E (the ratio of the separatrix length to the diameter) sufficiently large so that magnetic field curvature is negligible over the central portion of the separatrix. For such cases, assuming no plasma pressure outside the separatrix and an infinitely long flux conserver ofuniform radius, an axial equilibrium constraint (first(fd3e)ri=ve-dlblyr.B-apr2nersd) isro=bt1aine-dI-!X:-;, 0 PM 2 2 where P= plpM is the plasma pressure normalized to the maximum pressure at the field null and where Xs = rslrw is the ratio of midplane separatrlx radius to coil radius. The average beta condition ofEq. (1) is a stringent constraint on the FRC radial pressure profile. It has been found to be consistent with the available experimental data,2 and has often been used in equilibrium and transport studies. Various corrections to Eq. (1) have been derived to account for finite particle orbits,3 plasma pressure on open field lines,4 end mirrors,4,s and toroidal field.4 Magnetic field curvature may significantly modify Eq. ( 1) if PRC equilibria are produced with sufficiently small values of E. Such small values of E could result from separatrix length shrinkage,6 which occurs whenever energy losses exceed po10idal flux loss. Such short equilibria are currently being studied experimentally. In the VLX device7 PRC confinement in a coil ofaspect ratio (ratio ofuniform coil region to coil diameter) variable in the range 1-4 is being investigated, and in the FRX-C device8 short PRC equilibria with E in the range 3-4 have been recently obtained with a coil aspect ratio of2.6. Even shorter FRC equilibria may be obtained in a modified version of this device9 with a coil aspect ratio of about 2. II. EQUILIBRIUM COMPUTATIONS In this paper, we compute the effect offield curvature on PRC equilibria. Short equilibria were obtained in straight cylinder geometry (no end mirrors) by using the same code as in previous work.4 In each of the cases discussed in this paper, the equilibrium length was adjusted by changing the total toroidal current; Ps (the pressure at the separatrix normalized to the pressure at the field null) and Xs were held fixed by adjusting the pressure profile, p (l/J). For each equilibrium, the integral given in Eq. (1) was computed and the ratio (f3 )I (1 - ! x;) was obtained. These ratios are shown as functions ofE in Fig. 1for various values ofXs andps' The open circles correspond to x. = 0.4 and P. = 0.6 and the solid circles correspond to Xs = 0.6 and Ps = 0.3. Also shown with a triangle in Fig. 1 is the analytical spherical Hill's vortex equilibrium (P. = 0) which is a solution in a uniform flux conserverlO in the limit ofx. -.0. For this case, onehas (f3) = at E = 1. Theequilibria represented in Fig. 1 by open circles do not appear to map towards the Hill's vortex value, presumably because of the finite values of X s andPs .Fixing Ps and reducing E produced rather odd-looking equilibria with a lot of magnetofluid outside the separatrix. To see what happens when this fluid is removed, two additional equilibria (open circles with a bar) were computed with x. = 0.4 and P. = 0.1. These equilibria have lower values of (f3) than those with x. = 0.4 and P. = 0.6 and better extrapolate toward the Hill's vortex value. The results -(\l N'w f!. x t: Ii 0.5 V 4 E FIG. 1. Computed values of (fJ ) relative to elongated values as functions of . The open circles are forx. = 0.4and,8. = 0.6, the open circles with a bar are for x. = 0.4 and P, = 0.1, and the solid circles are for x. = 0.6 and ,8. = 0.3. The triangle is for the Hill's vortex case with x, =,8. = 0, and the solid line is the analytical expression of Eq. (3) with x. = 0.5. For each case, the value (fJ) u' the average of beta over the separatrix volume, normalized to I !X , is shown with a cross. 3711 Phys. Fluids 29 (11), November 1986 0031-9171/86/113711-04$01.90 @ 1986 American Institute of Physics 3711 FIG. 2. Computed radial pressure profiles relative to the maximum pressure PM for x, =0.4 and (a) E = 3.9, (b) E = 2.7, and Copyright E = 1.4. The dotted line is an approximate sharp-boundary model for curve Copyright. 1 (2) lrw B 2 -- ---21TrdrI . 11'':; 0 21l0PM z>O The first integral on the right-hand side of Eq. (2) can be evaluated using B 2/2JLOPM = I-p/PM forr rs ' In the second integral we use flux conservation, B Iz>o =Bw (1 - x;), and our modified radial pressure balance, B /2JLoPM = 1 - 8. With these substitutions, and some rearrangement, Eq. (2) can be written (f3) = l-x;12 - (8/2)(1-x;). (3) We now make an estimate of the quantity 8. With field-line curvature included, radial pressure balance, in the midplane, becomes ty is quite large and localized near the separatrix. This is a rather drastic approximation, but since the quantity ofinterest, (/3 ), is an integrated quantity, it is not as bad as might be expected. Note that as E becomes large, this model is trivially exact with {j = O. With this model the integral for (f3 ) in Eq. ( 1) can be evaluated, following the derivation outlined in previous work. 1 We start with Eq. (A7) of Ref. 1. When normalized to 11'':;PM' this equation becomes 1 lrw B 2 (f3) = - --211'7 dr I 11'':; 0 21l0PM z = 0 B 2 JB 2 p+-= --dr, 2110 lloP whereP is the radius ofcurvature of the magnetic field lines. To a good approximation, FRCseparatrices 's (z) have been shown experimentally2.4 to be elliptical in shape, so that at r = rs and z = 0 one obtains p = - rs c. Assuming that the integrated curvature factor {j is proportional to E- 2, we write 8 = a/c. The coefficient a can now be estimated by computing it for the simplest finite-length FRC equilibrium, the Hill's vortex. We do so by requiring Eq. (3) to hold for this equilibrium. With (/3) = 2/3, X s <1, and E = 1, we obtain {j = a = 2/3. We therefore assume {j = 2I3c and rewrite Eq. (3) as (/3) = 1 -x;12 - (1- x;)/3c. (4) The solid line on Fig. 1corresponds to Eq. (4) withxs = 0.5. This curve fits well the numerical data over the entire range of E. Equation (4) provides a simple approximation to (f3) for arbitrary aspect ratio E that can easily be evaluated experimentally. The factor 1 - x; that appears in Eq. (4) implies decreased curvature effects with larger values ofxs ' in qualitative agreement with the numerical data ofFig. 1. This can probably be attributed to the effect ofthe flux conserving wall; at large xs ' the plasma is pushed against the straight wall, and tends to become flat. The numerical data of Fig. 1 also suggest that curvature effects are lessened with increasing/ 3s; this effect is not included in Eq. (4). The FRG total temperature T is commonly inferred from radial pressure balance once B w at radius r = rw and nM at radius, = 's/21/2 are determined from excluded flux measurements and from side-on interferometry. Therefore, the expressions PM = nMT= B l2llo(1-{j) and {j = 21 o in Fig. I show that curvature effects are negligible for E 4 but rapidly become significant as E approaches unity. Comparison between the various cases suggests that curvature effects are somewhat reduced by larger values ofXs and /3s . For highly elongated FRC's, it is normally assumed that (f3 ), the midplane-averaged beta, is the same as (f3) v' the volume-averaged beta. Because (f3) is easily determined experimentally, it is often used, for the case ofhighly elongated FRC's, as a substitute for the more physical quantity (f3 ) v However, as E approaches 1, the values of (f3) v can diverge significantly from the values of(f3 ). This is illustrated in Fig. 1 where the values of (f3) v' normalized to 1 - x;, are shown with crosses for each computed case. Note that both (/3) and (/3 ) v decrease with decreasing E. However, the ratio (f3 ) v / (f3 ) decreases from O.95 to 0.69 as E decreases from 6 to 1. Hence, there is no obvious relation between (f3 ) and (/3 ) v ; furthermore. because ofpossible wide variations in the axial distribution of flux surfaces with variations inP ( t/!),4a simple relation between (f3 ) and (f3 )" may be quite difficult to find. III. ANALYTIC MODEL In order to gain more insight, an attempt was made to fit a simple analytical model for curvature effects. Thenumerical equilibria revealed midplane radial profiles of total pressure (p +B 2/2JLO) with a drop largely localized in the vicinity of the separatrix. Three examples are given in Fig. 2 for various values ofE and for Xs = 0.4. These examples suggest an approximate model with uniform regions oftotal pressure inside and outside the separatrix. An example for curve Copyright is sketched with a dotted line on Fig. 2. We assumeP +B 2/ 2JLo =PM inside the separatrix andp +B 2/2p,0 = PM (I - {j) outside the separatrix. Note that the discontinuity in total pressure in this model implies that B VB is proportional to a delta function at the separatrix, i.e., implies that this quanti- (a) lb) a. ..... 0 ::t. (e) C\l -. C\I CO as + eo 3712 Phys. Fluids, Vol. 29, No. 11, November 1986 M. Tuszewski and R. L. Spencer 3712 (5) ACKNOWLEDGMENT This research was supported by the United States Department of Energy. FIG. 3. Computed values ofs for short equilibria, normalized to the corresponding values off.> 1 for highly elongated equilibria (E = 6.3 for x, = 0.4 and E = 4.5 for x, = 0.6) are displayed as functions of E. The various cases are the same as those of Fig. 1. The solid lines correspond to the relation s-;- [I +61X:(l - 6) ]1/2, normalized to the same expression evaluated forE = 6.3 (x, = 0.4, uppercurve) and forE = 4.5 (x, = 0.6, lowercurve). 4 5 6 E 2 3 With 8 = 213 , this relation is shown with solid lines in Fig. 3 for Xs = 0.4 and for Xs = 0.6. The quantity 8,,> I is a value of8 corresponding to E = 6.3 for X s = 0.4 and to E = 4.5 for X s = 0.6. We observe from Fig. 3 that the numerical values ofnormalized Svalues are indeed in good agreement with the approximate upper-bound scaling. The relation (J-Bwx; [1 +l5/x;(1 - tJ>] 1/2 (6) may therefore be a useful way to use simple magnetic measurements in present experiments to estimate the flux confinement times ofshort FRC's. Similarly, the values of (f3 >" shown in Fig. I could perhaps be used to estimate the energy confinement times of short FRC's from the relation E- (f3 >v VB /(l -15),whereEistheplasmaenergyand V is the separatrix volume. IV. CONCLUSIONS In conclusion, some features of short FRC equilibria have been studied numerically. Specifically, it is shown that magnetic field curvature significantly increases the FRC magnetization for elongations E<4. This should improve FRC transport and might decrease the effectiveness ofkinetic stabilization. The transport parameter (f3 >is computed and a simple analytical approximation for it is derived. The relative increases ofthe stability parametersand ofthe FRC trapped flux (J over one-dimensional estimates are evaluated. It is found that curvature effects are somewhat reduced by increased values of X s and f3s. ...... len len 1.5 3 provide a temperature estimate for cases with low values ofE. The values of (f3 >given by Eq. (4) can be used to extend existing one-dimensional transport models, such as the one of Ref. 3, to the range E-2-4. In this range, end effects that reducell.'2confinementtimesbyafactor /(+ Vremain small while the decreased (f3 > (and therefore decreased pressure gradients) should yield improved cross-field transport. A parameter of importance for FRC stability is -s= ir'r-d-r, R rsPi the approximate number13 of ion gyroradii, Pi' between the field null R and the separatrix rs A recent kinetic theory14 of the internal tilt mode suggests sufficiently small growth rates for present experiments with $< 1-2 to be stable, but observable instability in future experiments with 8> 3-4. If 8 has the same significance with regard to stability for these short equilibria, then such equilibria might represent a simpler way to study the tilt mode. Their shorter length, however, could substantially alter the eigenfunction (for instance, by changing it from an axial displacement to a rigid rotation, or by changing the important instability from an internal tilt mode to an external tilt mode, or by changing it into some combination of all of these) making it difficult to compare with the physics oflong equilibria. For cases with small values ofE, the usual one-dimensional estimates'2 of 8 become inaccurate. Since 8 is proportional to (J/rs , where (J is the FRC poloidal fiux, and where we have assumed constant ion temperature, one can readily compute the relative increase of 8 over its elongated value, for a given value of X s ' by calculating (J ratios. The ratios S/8,,>1 = (J/(J,,> I from the numerical data of Fig. 1 are shown in Fig. 3. For the cases with Xs = 0.4 we used an equilibrium with E = 6.3 to obtain (J,,> I' and for Xs = 0.6 we used an equilibrium with E= 4.5. We observe from Fig. 3 significant relative increases of s for E<3 that correspond to the enhanced magnetization suggested in Fig. I by lower values of (f3 ). The reduction of curvature effects with larger values ofX s and f3s noted previously is also apparent in Fig. 3. For elongated FRC's, Eq. (1) has been usedl 6 to provide bounds on the FRC poloidal flux (J. For short FRC's, Eq. (4) can be used in a similar way to obtain I.:B X (l ) 11' III "'4 + x;(1 _ tJ) x; ( tJ )112 I = [1 +tJ">I/X;O - tJ">l >] 1/2 . 3713 Phys. Fluids, Vol. 29, No. 11, November 1986 M. Tuszewski and R. L. Spencer 3713 IW. T. Armstrong, R. K. Linford, J. Lipson, D. A. J1latts, and E. G. Sherwood, Phys. Fluids 24, 2068 (1981). 2M. Tuszewski, Plasma Phys. 26, 991 (1984). 3M. Tuszewski and R. K. Linford, Phys. Fluids 25, 765 (1982). 4R. L. Spencer and M. Tuszewski, Phys. Fluids 28, 1810 (1985). 'K. Suzuki and S. Hamada, J. Phys. Soc. Jpn. 53,16 (1984). 6M. Tuszewski, W. T. Armstrong, R. R. Bartsch, R. E. Chrien, J.Copyright Cochrane, Jr., R. W. Kewish, Jr., P. K1ingner, R. K. Linford, K. F. McKenna, D. J. Rej, E. G. Sherwood, and R. E. Siemon, Phys. Fluids 25, 1696 (1982). 'J. A. Wilson, Z. A. Pietrzyk, and D. E. Lotz, Bull. Am. Phys. Soc. 29, 1360 (1984). 8M. Tuszewski, in Proceedings of the 7th Symposium on the Physics and Technology ofCompact Toroids, Santa Fe, 1985 (Los Alamos National Laboratory, Los Alamos, NM, 1986), p. 142. . Tuszewski, R. E. Chrien, E. J. Yavomik, W. T. Armstrong, W. Hugrass, R. K. Linford, K. F. McKenna, D. J. Rej, and R. E. Siemon, in Proceedings ofthe 7th Symposium on the Phpics and Technology ofCompact Toroids, Santa Fe, 1985 (Los Alamos National Laboratory, Los Alamos, NM, 1986), p. IS2. 10Jt. L. Spencer and D. W. Hewett, Phys. Fluids 15, 1365 (1982). I1S. P. Auerbach and W.Copyright Condit, Nucl. Fusion 21, 927 (1981). 12K. Nguyen and T. Kammash, Plasma Phys. 24,177 (1982). 13J. T. Slough, A. L. Hotrman, R. D. Milroy, D. G. Harding, and L.Copyright Steinhauer, Nucl. Fusion 24, 1537 (1984). 1 4J. L. Schwarzmeier, H. R. Lewis, D.Copyright Barnes, and C. E. Seyler, Bull. Am. Phys. Soc. 29,1327 (1984). 3714 Phys. Fluids, Vol. 29. No. 11, November 1986 M. Tuszewski and R. L. Spencer 3714